422edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
| {{monzo| 669 -422 }}
| {{monzo| 669 -422 }}
| {{mapping| 422 669 }}
| {{mapping| 422 669 }}
| -0.1308
| &minus;0.1308
| 0.1308
| 0.1308
| 4.60
| 4.60
Line 32: Line 24:
| {{monzo| 23 6 -14 }}, {{monzo| 33 -34 9 }}
| {{monzo| 23 6 -14 }}, {{monzo| 33 -34 9 }}
| {{mapping| 422 669 980 }}
| {{mapping| 422 669 980 }}
| -0.1469
| &minus;0.1469
| 0.1092
| 0.1092
| 3.84
| 3.84
Line 39: Line 31:
| 4375/4374, 589824/588245, 29360128/29296875
| 4375/4374, 589824/588245, 29360128/29296875
| {{mapping| 422 669 980 1185 }}
| {{mapping| 422 669 980 1185 }}
| -0.1852
| &minus;0.1852
| 0.1155
| 0.1155
| 4.06
| 4.06
Line 46: Line 38:
| 3025/3024, 4375/4374, 5632/5625, 589824/588245
| 3025/3024, 4375/4374, 5632/5625, 589824/588245
| {{mapping| 422 669 980 1185 1460 }}
| {{mapping| 422 669 980 1185 1460 }}
| -0.1679
| &minus;0.1679
| 0.1090
| 0.1090
| 3.83
| 3.83
Line 53: Line 45:
| 1716/1715, 2080/2079, 2200/2197, 3025/3024, 5632/5625
| 1716/1715, 2080/2079, 2200/2197, 3025/3024, 5632/5625
| {{mapping| 422 669 980 1185 1460 1562 }}
| {{mapping| 422 669 980 1185 1460 1562 }}
| -0.1930
| &minus;0.1930
| 0.1142
| 0.1142
| 4.02
| 4.02
Line 60: Line 52:
| 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2200/2197, 2431/2430
| 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2200/2197, 2431/2430
| {{mapping| 422 669 980 1185 1460 1562 1725 }}
| {{mapping| 422 669 980 1185 1460 1562 1725 }}
| -0.1744
| &minus;0.1744
| 0.1151
| 0.1151
| 4.05
| 4.05
Line 67: Line 59:
| 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1716/1715, 2200/2197
| 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1716/1715, 2200/2197
| {{mapping| 422 669 980 1185 1460 1562 1725 1793 }}
| {{mapping| 422 669 980 1185 1460 1562 1725 1793 }}
| -0.1839
| &minus;0.1839
| 0.1106
| 0.1106
| 3.89
| 3.89
Line 74: Line 66:
| 736/735, 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1496/1495, 1716/1715
| 736/735, 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1496/1495, 1716/1715
| {{mapping| 422 669 980 1185 1460 1562 1725 1793 1909 }}
| {{mapping| 422 669 980 1185 1460 1562 1725 1793 1909 }}
| -0.1675
| &minus;0.1675
| 0.1142
| 0.1142
| 4.02
| 4.02
|}
{{comma basis end}}
* 422et has lower absolute errors than any previous equal temperaments in the 17-, 19- and 23-limit. In the 17- and 19-limit it beats [[400edo|400]] and is bettered by [[460edo|460]]. In the 23-limit it beats [[373edo|373g]] and is bettered by [[525edo|525]].  
* 422et has lower absolute errors than any previous equal temperaments in the 17-, 19-, and 23-limit. In the 17- and 19-limit it beats [[400edo|400]] and is bettered by [[460edo|460]]. In the 23-limit it beats [[373edo|373g]] and is bettered by [[525edo|525]].  


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
Line 126: Line 112:
|-
|-
| 2
| 2
| 153\422<br>(58\422)
| 153\422<br />(58\422)
| 435.07<br>(164.93)
| 435.07<br />(164.93)
| 9/7<br>(11/10)
| 9/7<br />(11/10)
| [[Semisupermajor]]
| [[Semisupermajor]]
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}


[[Category:Countritonic]]
[[Category:Gamera]]
[[Category:Gamera]]
[[Category:Vishnu]]
[[Category:Vishnu]]
[[Category:Countritonic]]

Revision as of 05:07, 16 November 2024

← 421edo 422edo 423edo →
Prime factorization 2 × 211
Step size 2.8436 ¢ 
Fifth 247\422 (702.37 ¢)
Semitones (A1:m2) 41:31 (116.6 ¢ : 88.15 ¢)
Consistency limit 27
Distinct consistency limit 27

Template:EDO intro

Theory

422edo is a zeta peak edo, though not zeta integral nor zeta gap. It is distinctly consistent through the 27-odd-limit, with harmonics of 3 through 23 all tuned sharp. The equal temperament tempers out the vishnuzma, [23 6 -14⟩ and the countritonic comma, [33 -34 9⟩, in the 5-limit; 4375/4374 and 589824/588245 in the 7-limit; 3025/3024, 5632/5625, and 9801/9800 in the 11-limit; 1716/1715, 2080/2079, and 2200/2197 in the 13-limit; 1156/1155, 1275/1274, and 2431/2430 in the 17-limit; 1216/1215, 1331/1330, 1445/1444, and 2432/2431 in the 19-limit; and 736/735, 1496/1495, and 1863/1862 in the 23-limit. It supports provides the optimal patent vals for gamera in the 7-limit and hemigamera in the 13-limit. Other notable temperaments it supports are vishnu, semisupermajor, and countritonic.

Prime harmonics

Approximation of prime harmonics in 422edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.41 +0.42 +0.84 +0.34 +1.18 +0.26 +1.07 +0.16 -0.19 +0.94
Relative (%) +0.0 +14.6 +14.6 +29.6 +12.0 +41.4 +9.1 +37.5 +5.7 -6.8 +32.9
Steps
(reduced) 		422
(0) 		669
(247) 		980
(136) 		1185
(341) 		1460
(194) 		1562
(296) 		1725
(37) 		1793
(105) 		1909
(221) 		2050
(362) 		2091
(403)

Subsets and supersets

Since 422 factors into 2 × 211, 422edo has subset edos 2edo and 211edo.

Regular temperament properties

Template:Comma basis begin |- | 2.3 | [669 -422⟩ | [⟨422 669]] | −0.1308 | 0.1308 | 4.60 |- | 2.3.5 | [23 6 -14⟩, [33 -34 9⟩ | [⟨422 669 980]] | −0.1469 | 0.1092 | 3.84 |- | 2.3.5.7 | 4375/4374, 589824/588245, 29360128/29296875 | [⟨422 669 980 1185]] | −0.1852 | 0.1155 | 4.06 |- | 2.3.5.7.11 | 3025/3024, 4375/4374, 5632/5625, 589824/588245 | [⟨422 669 980 1185 1460]] | −0.1679 | 0.1090 | 3.83 |- | 2.3.5.7.11.13 | 1716/1715, 2080/2079, 2200/2197, 3025/3024, 5632/5625 | [⟨422 669 980 1185 1460 1562]] | −0.1930 | 0.1142 | 4.02 |- | 2.3.5.7.11.13.17 | 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2200/2197, 2431/2430 | [⟨422 669 980 1185 1460 1562 1725]] | −0.1744 | 0.1151 | 4.05 |- | 2.3.5.7.11.13.17.19 | 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1716/1715, 2200/2197 | [⟨422 669 980 1185 1460 1562 1725 1793]] | −0.1839 | 0.1106 | 3.89 |- | 2.3.5.7.11.13.17.19.23 | 736/735, 1156/1155, 1216/1215, 1275/1274, 1331/1330, 1445/1444, 1496/1495, 1716/1715 | [⟨422 669 980 1185 1460 1562 1725 1793 1909]] | −0.1675 | 0.1142 | 4.02 Template:Comma basis end

  • 422et has lower absolute errors than any previous equal temperaments in the 17-, 19-, and 23-limit. In the 17- and 19-limit it beats 400 and is bettered by 460. In the 23-limit it beats 373g and is bettered by 525.

Rank-2 temperaments

Template:Rank-2 begin |- | 1 | 81\422 | 230.33 | 8/7 | Gamera |- | 1 | 111\422 | 315.64 | 6/5 | Egads |- | 1 | 153\422 | 435.07 | 9/7 | Supermajor |- | 1 | 207\422 | 588.63 | 128/91 | Countritonic |- | 2 | 25\422 | 71.09 | 25/24 | Vishnu / acyuta |- | 2 | 81\422 | 230.33 | 8/7 | Hemigamera |- | 2 | 153\422
(58\422) | 435.07
(164.93) | 9/7
(11/10) | Semisupermajor Template:Rank-2 end Template:Orf

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